Implicit Differentiation

By SkytrailGroup Mathematics Editorial Team April 12, 2026

Method

Differentiate both sides with respect to x, applying chain rule to y terms (treating y as a function of x), then solve for dy/dx.

Examples

Example 1. Find dy/dx for x²+y²=25.

Solution. 2x+2y(dy/dx)=0 → dy/dx=−x/y.

In Depth

Implicit differentiation finds \(dy/dx\) when \(y\) is defined implicitly by an equation \(F(x,y)=0\), without solving for \(y\) explicitly. Differentiate both sides with respect to \(x\), treating \(y\) as a function of \(x\) and applying the chain rule to every \(y\)-term.

The implicit function theorem gives conditions under which an equation \(F(x,y)=0\) defines \(y\) as a function of \(x\) near a point: if \(F\) is continuously differentiable and \(\partial F/\partial y \neq 0\) at the point, then \(y\) is locally a function of \(x\) with \(dy/dx = -(\partial F/\partial x)/(\partial F/\partial y)\).

In multivariable calculus, implicit differentiation extends to surfaces \(F(x,y,z)=0\): the gradient \(\nabla F\) is normal to the surface, and partial derivatives of \(z\) with respect to \(x\) and \(y\) are \(\partial z/\partial x = -(F_x/F_z)\) and \(\partial z/\partial y = -(F_y/F_z)\).

Applications: finding tangent lines to curves defined by polynomial equations (algebraic curves), computing derivatives of inverse functions (\(d/dx[\arcsin x] = 1/\sqrt{1-x^2}\) is derived by implicit differentiation of \(\sin y = x\)), and analyzing level curves in optimization.

Key Properties & Applications

Implicit differentiation applies the chain rule to equations \(F(x,y)=0\) without solving for \(y\) explicitly. Differentiating both sides with respect to \(x\) and treating \(y\) as a function of \(x\) gives \(dy/dx=-F_x/F_y\) (implicit function theorem formula).

The implicit function theorem guarantees that \(F(x,y)=0\) defines \(y\) as a smooth function of \(x\) near a point \((x_0,y_0)\) where \(F_y\neq0\). In higher dimensions, \(F(\mathbf{x},\mathbf{y})=\mathbf{0}\) defines \(\mathbf{y}\) as a function of \(\mathbf{x}\) when the Jacobian \(\partial F/\partial\mathbf{y}\) is invertible.

Applications: finding tangent lines to curves defined implicitly (circles, ellipses, folium of Descartes); related rates problems (how fast is the area of a circle changing as the radius changes?); and computing derivatives of inverse functions (\(d/dx[\arcsin x]=1/\sqrt{1-x^2}\) via implicit differentiation of \(\sin y=x\)).